1. Field of the Invention
The present invention relates generally to modeling the concentrations and mobilities of electrons and holes in semiconductors, from Hall and resistivity data. More particularly, the invention relates to fully automated modeling that does not require making assumptions about the number of types of carriers in the semiconductor.
2. Description of the Related Art
Mixed-conduction effects (that is, conduction that is attributable to several types of electrons and holes) quite often have a strong influence on the magneto-transport properties of semiconductor materials, including bulk samples, thin films, quantum wells, and processed devices. Multiple species, due to majority and minority carriers in the active region, intentional n and p doping regions (as well as unintentional doping non-uniformities along the growth axis), localization in multiple active regions, carriers populating buffer layers and substrates, 2D populations at surfaces and interface layers, and carriers populating different conduction band minima or valence band maxima (e.g., .GAMMA., X, and L valley electrons) all tend to contribute simultaneously to the conduction in real materials of interest to industrial characterization and process control, and to research investigations of novel materials and phenomena. Standard measurements of the resistivity and Hall coefficient at a single magnetic field are of limited use when applied to systems with prominent mixed-conduction, since they provide only averaged values of the carrier concentration and mobility, which are not necessarily representative of any of the individual species. Far more information becomes available if one performs the magneto-transport experiments as a function of magnetic field, because in principle one can then deconvolve the data to obtain densities and mobilities for each type of carrier present.
A representative sample exhibiting the Hall effect is shown in FIG. 1, with the coordinate axes labeled. Coordinate axes are used herein for illustration purposes. The motion of carriers in an isotropic sample exhibiting the Hall effect may be described by: EQU J.sub.x =.sigma..sub.xx E.sub.x +.sigma..sub.xy E.sub.y EQU J.sub.y =.sigma..sub.yx E.sub.x +.sigma..sub.yy E.sub.y (1)
where J is the current density in the x or y direction, where .sigma..sub.xx is the diagonal conductivity, or the conductivity in the x direction where the electric field E is applied in the x direction, where .sigma..sub.xy is the Hall conductivity, or the conductivity in the y direction where the electric field is applied in the x direction and a magnetic field B is applied in the z direction, and where E.sub.x and E.sub.y are the x and y components of E. For isotropic materials .sigma..sub.xx =.sigma..sub.yy and .sigma..sub.yx =-.sigma..sub.xy.
The experimental Hall coefficient R.sub.H and resistivity .rho. are related to the components of the conductivity tensor by the expressions ##EQU1## and ##EQU2## For a sample containing more than one type of carrier, the conductivity tensor components can be expressed as a sum over the m species present within the multi-carrier system: ##EQU3## and ##EQU4## where B is the magnetic field applied along the z axis, n.sub.i and .mu..sub.i are the concentration and mobility of the ith carrier species, e is the per carrier charge of 1.6.times.10.sup.-19 coulombs, and S.sub.i is the charge sign of the ith carrier species (+1 for holes and -1 for electrons). It is primarily the (1+.mu..sub.i.sup.2 B.sup.2) terms in the denominators which differentiate the contributions by the various carrier species. The contributions due to higher-mobility carriers are the first to be "quenched" as B is increased, i.e., a given species exerts far less influence on R.sub.H (B) and .rho.(B) once .mu..sub.i B&gt;&gt;1. This phenomenon provides the field-dependent Hall data with their high degree of sensitivity to the individual mobilities.
Several techniques are known for analyzing magnetic field dependent Hall data to model carrier concentrations. The traditional technique for analyzing magnetic-field-dependent Hall data is the Multi-Carrier Fitting (MCF) procedure, whereby equations (2) and (3) above are employed to fit experimental data. In this method, n.sub.i and .mu..sub.i are the fitting parameters and the number of carriers m is typically between 1 and 5. A significant drawback of the MCF is its arbitrariness. One must not only make prior assumptions about the approximate densities and mobilities of the various electron and hole species, but a decision must also be made in advance with respect to how many carriers of each type of assume. Injudicious guesses can lead to misleading or ambiguous results, and the fit to the experimental Hall data is not unique. A second disadvantage is that since discrete "delta function" mobilities are assumed, the fit yields no information about the "linewidth" of each mobility feature.
In order to overcome these shortcomings, Beck and Anderson (W. A. Beck and J. R. Anderson, J. Appl. Phys. 62, 541 (1987)) (BA) proposed an approach known as the Mobility Spectrum Analysis, in which an envelope of the maximum conductivity is determined as a continuous function of mobility. Equations (2) and (3) are rewritten in integral form ##EQU5## and ##EQU6## where s.sup.p and s.sup.n are the hole and electron conductivity density functions (i.e., the conductivity associated with a concentration of carriers at each mobility, also referred to herein as the mobility spectra). These are given by: EQU s.sup.p (.mu.)=p(.mu.)e.mu., EQU s.sup.n (.mu.)=n(.mu.)e.mu. (6)
where p(.mu.) and n(n) are the hole and electron density functions (i.e., the concentration of carriers at each mobility). However, while the goal is to find s.sup.p (.mu.) and s.sup.n (.mu.), these are not uniquely defined by the measured .sigma..sub.xx (B) and .sigma..sub.xy (B). Given values for the conductivity tensor at N different magnetic fields define a 2N-dimensional space which has, at most, 2N independent basis vectors. Since equations (4) and (5) represent an expansion of the data in terms of an infinite basis, the expansion cannot be unique. Using a rather complex mathematical formalism, BA instead obtained unique envelopes s.sub.BA.sup.n (.mu.) and s.sub.BA.sup.p (.mu.) which represent physical .delta.-like amplitudes at .mu.. While this is not as valuable as finding unique s.sup.n (.mu.) and s.sup.p (.mu.), it is still useful in that the most prominent carrier species may usually be identified from the peaks in the envelope spectrum. A major advantage of the mobility spectrum analysis over the MCF is that it is non-arbitrary, i.e., no prior assumptions are required. It is also computer automated, and provides a visually meaningful output format. However, the significant disadvantage is that the information obtained is primarily qualitative rather than quantitative, since one does not actually obtain a fit to the experimental Hall and resistivity data.
In 1988, Meyer et al. (J. R. Meyer, C. A. Hoffman, F. J. Bartoli, D. J. Arnold, S. Sivananthan, and J. P. Faurie, Semicond. Sci. Technol. 8, 805 (1993); C. A. Hoffman, J. R. Meyer, F. J. Bartoli, J. W. Han, J. W. Cook, Jr., and J. F. Schetzina, Phys, Rev. B 40, 3867 (1989)) developed a Hybrid Mixed Conduction Analysis (HMCA), which was extensively tested between 1988 and 1994. In that method, the BA mobility spectrum is used to determine the number of carrier species and roughly estimate their densities and mobilities, and the MCF is then used to obtain a final quantitative fit. However, a degree of arbitrariness is still present in making decisions based on the BA spectra, the multi-carrier fits do not always converge (e.g., the "best" fit may be obtained with one of the species having a density approaching infinity and a mobility approaching zero), and again one obtains no linewidth information, since the results are expressed only in terms of a small number of species with discrete mobilities. These difficulties effectively preclude full computer automation of some stages of the procedure. Brugger and Koser (H. Brugger and H. Koser, III-V Reviews 8, 41 (1995)) have more recently discussed a similar approach, which suffers from the same limitations.
In 1991-92, Dziuba and Gorska (Z. Dziuba and M. Gorska, J. Phys. III France 2, 99 (1992); Z. Dziuba, Acta Physica Polonica A 80, 827 (1991)) (DG) discussed a technique that was more ambitious than that of BA, namely to quantitatively derive the actual mobility distribution instead of just an upper-bound envelope. In their approach, the integrals appearing in the conductivity tensor expression of Eqs. (4) and (5) are approximated by sums of the partial contributions by carriers having a grid of discrete mobilities. ##EQU7## and ##EQU8## where s.sup.p (.mu..sub.i) and s.sup.n (.mu..sub.i) are hole and electron conductivities associated with the discrete mobility grid point .mu..sub.i. The parameter N defines both the number of points i in the final mobility spectrum and the number of magnetic fields j at which pseudo-data points .sigma..sub.xx (B.sub.j) and .sigma..sub.xy (B.sub.j), (which are denoted in what follows .sigma..sub.xx.sup.j (exp) and .sigma..sub.xy.sup.j (exp)) are satisfied by the model. Interpolation is used to obtain the pseudo-data points from the actual experimental data, .sigma..sub.xx.sup.exp and .sigma..sub.xy.sup.exp, which are usually measured at a much smaller number of fields B.sub.k. Using an initial trial spectrum, DG solved the set of equations (7) and (8) using the Jacobi iterative procedure, in which the transformation matrix elements 1/(1+.mu..sub.i.sup.2 B.sub.i.sup.2) and .mu..sub.i B.sub.i /(1+.mu..sub.i.sup.2 B.sub.i.sup.2) are simplified because of the specific choice of mobility points (.mu..sub.i =1/B.sub.i) in the s.sup.p (.mu..sub.i) and s.sup.n (.mu..sub.i) spectra. An important consequence of this choice of mobility points is that the mobility range is limited to .mu..sub.min .ltoreq..mu..ltoreq..mu..sub.max. Here .mu..sub.min =1/B.sub.max.sup.exp and .mu..sub.max =1/B.sub.min.sup.exp, where B.sub.max.sup.exp and B.sub.min.sup.exp are the minimum and maximum experimental magnetic fields. The goal of the procedure is to find those 2N variables s.sup.p (.mu..sub.i) and s.sup.n (.mu..sub.i) which solve the 2N equations in the system represented by Eqs. (7) and (8). In general, "non-physical" negative values of s.sup.p (.mu..sub.i) and s.sup.n (.mu..sub.i) are obtained for some regions of the spectra.
The original version of the Quantitative Mobility Spectrum Analysis (U.S. Pat. No. 5,789,931) (o-QMSA), which was developed in 1994 and 1995, was the first fully automated algorithm to combine the quantitative accuracy of the conventional least-squares MCF result with a visually meaningful mobility spectrum output format. An important difference between o-QMSA and DG is that the number of variables in o-QMSA is significantly smaller than the number of quasi-data points. The objective of the o-QMSA algorithm was thus not to reproduce the data exactly, but to obtain a spectrum which best fits the data to the extent allowed by the constricted number of variables.
In one embodiment of o-QMSA, an iteration procedure analogous to that used by DG is employed, except that s.sup.n (.mu.) and s.sup.p (.mu.) are both constrained to be non-negative at all iteration steps, which corresponds to a requirement that no carriers can contribute negative conductivities. The key consequence is that by forcing s.sup.n (.mu..sub.i)&gt;0 and s.sup.p (.mu..sub.i)&gt;0, many of the "variables" are no longer varied in any given fit, and the 2N data points must be fit using considerably less than 2N parameters. Extensive testing confirmed that the imposition of this condition removes the inherent instability of the DG procedure.
The Gauss-Seidel iterative approach with Successive Over-relaxation was used to solve the conductivity tensor equations for s.sup.p and s.sup.n. Coefficients .omega..sub.x and .omega..sub.y were employed to control the speed of convergence, with preferred values that minimize the net error in the fits being .omega..sub.x =0.03 and .omega..sub.y =0.003.
The o-QMSA approach was extensively tested on resistivity and Hall data as a function of magnetic field (typically 0-7 T) and temperature for diverse types of semiconductor samples. Although the advantages of the o-QMSA over all previous mixed-conduction techniques were clear, several drawbacks also came to light. First, although reasonable fits to the experimental conductivity tensor could nearly always be obtained, the error was often considerably larger than that obtained using the MCF. Secondly, when the procedure was extended to a large number of iterations, the spectra often tended to collapse to a collection of discrete delta-function-like features, i.e. the linewidth information in the Hall data could not reliably be extracted. Thirdly, peaks corresponding to low-mobility features (having .mu.&lt;B.sub.max.sup.-1) were generated by extrapolating the data to magnetic fields B more than an order of magnitude beyond the experimental range bounded by B.sub.max.sup.exp. This procedure yields surprisingly reasonable results, bit is of questionable validity. And finally, the o-QMSA approach tends to produce a large number of "ghost" peaks, i.e. unphysical low-density minority features that can rob carriers from the majority peaks and thereby distort the derived carrier concentrations.